$X_{2(g)} + Y_{2(g)} \rightleftharpoons 2Z_{(g)}$
$X_{2(g)}$ and $Y_{2(g)}$ are added to a $1 \ L$ flask and it is found that the system attains the above equilibrium at $T \ K$ with the number of moles of $X_{2(g)}$, $Y_{2(g)}$ and $Z_{(g)}$ being $3$, $3$ and $9 \ mol$ respectively (equilibrium moles). Under these conditions of equilibrium, $10 \ mol$ of $Z_{(g)}$ is added to the flask and the temperature is maintained at $T \ K$. Then the number of moles of $Z_{(g)}$ in the flask when the new equilibrium is established is . . . . . . . (Nearest integer).

For the reaction $X(s) \rightleftharpoons Y(s) + Z(g)$,the plot of $\ln \frac{p_z}{p^\ominus}$ versus $\frac{10^4}{T}$ is given below,where $p_z$ is the pressure (in bar) of the gas $Z$ at temperature $T$ and $p^\ominus = 1 \ bar$.
(Given,$\frac{d(\ln K)}{d(\frac{1}{T})} = -\frac{\Delta H^\ominus}{R}$,where the equilibrium constant,$K = \frac{p_z}{p^\ominus}$ and the gas constant,$R = 8.314 \ J \ K^{-1} \ mol^{-1}$)
$(1)$ The value of standard enthalpy,$\Delta H^\ominus$ (in $kJ \ mol^{-1}$) for the reaction is. . . . . . .
$(2)$ The value of $\Delta S^\ominus$ (in $J \ K^{-1} \ mol^{-1}$) for the given reaction,at $1000 \ K$ is. . . . . .
Give the answer for $(1)$ and $(2)$

$37.8 \ g$ $N_2O_5$ was taken in a $1 \ L$ reaction vessel and allowed to undergo the following reaction at $500 \ K$:
$2N_2O_{5(g)} \rightarrow 2N_2O_{4(g)} + O_{2(g)}$
The total pressure at equilibrium was found to be $18.65 \ bar$. Then,$K_p = \text{ . . . . . . } \times 10^{-2}$ [nearest integer].
Assume $N_2O_5$ to behave ideally under these conditions.
Given: $R = 0.082 \ bar \ L \ mol^{-1} \ K^{-1}$

For the reversible reaction $2NO_2 \leftrightarrow[K_2]{K_1} N_2O_4$,the rate of disappearance of $NO_2$ is given by:

Consider the following gas phase reaction being carried out in a closed vessel at $25^\circ\text{C}$: $2A(g) \rightarrow 4B(g) + C(g)$. The table provides the total pressure of the system at different time intervals. Calculate the pressure of $C(g)$ at $30$ minutes time interval.

Time (min)	Total pressure (mm Hg)
$30$	$300$
$\infty$	$600$

$56 \, g$ of nitrogen and $8 \, g$ of hydrogen gas are heated in a closed vessel. At equilibrium,$34 \, g$ of ammonia are present. The equilibrium number of moles of nitrogen,hydrogen,and ammonia are respectively: